Effect of annealing on the magnetic microstructure of high-pressure torsion iron: the relevance of higher-order contributions to the magnetic small-angle neutron scattering cross section

The role of higher-order small-angle neutron scattering effects in ultra-fine-grained pure iron prepared by combining high-pressure torsion with a post-annealing process is investigated.

The development of higher-order micromagnetic small-angle neutron scattering theory in nanocrystalline materials is still in its infancy. One key challenge remaining in this field is understanding the role played by the microstructure on the magnitude and sign of the higher-order scattering contribution recently observed in nanocrystalline materials prepared by high-pressure torsion. By combining structural and magnetic characterization techniques, namely X-ray diffraction, electron backscattered diffraction and magnetometry with magnetic small-angle neutron scattering, this work discusses the relevance of higher-order terms in the magnetic small-angle neutron scattering cross section of pure iron prepared by high-pressure torsion associated with a post-annealing process. The structural analysis confirms: (i) the preparation of ultra-fine-grained pure iron with a crystallite size below 100 nm and (ii) rapid grain growth with increasing annealing temperature. The analysis of neutron data based on the micromagnetic small-angle neutron scattering theory extended to textured ferromagnets yields uniaxial magnetic anisotropy values that are larger than the magnetocrystalline value reported for bulk iron, supporting the existence of induced magnetoelastic anisotropy in the mechanically deformed samples. Furthermore, the neutron data analysis revealed unambiguously the presence of non-negligible higher-order scattering contributions in high-pressure torsion iron. Though the sign of the higher-order contribution might be related to the amplitude of the anisotropy inhomogeneities, its magnitude appears to be clearly correlated to the changes in the microstructure (density and/or shape of the defects) induced by combining high-pressure torsion and a post-annealing treatment.
plastic strain uniformly irrespective of the grain orientation. Although the characteristic microstructures produced by HPT can improve the mechanical properties, a considerable effort is devoted to understanding the effect of HPT on the magnetic properties. HPT already represents a promising approach to control the macroscopic magnetic properties of metals, in particular it has become possible to induce ferromagnetism in paramagnetic metals (Cepeda-Jimé nez et al., 2016 or to improve their soft-magnetic character (Scheriau et al., 2010). More recently, magnetic small-angle neutron scattering (SANS) experiments reported a significant increase of the effective magnetic anisotropy in HPT Fe (Oba et al., 2020) and HPT Ni Zaporozhets et al., 2022), opening up a new route for the development of advanced magnetic materials using severe plastic-deformation techniques (Horita & Edalati, 2020). However, the analysis of the magnetic SANS cross section in HPT Ni also revealed an unusual, predominant longitudinal sine-squared-type angular anisotropy, which is beyond the classical second-order micromagnetic SANS theory and can be qualitatively understood by the higher-order theory . A quantitative analysis of such a dominant (negative) higher-order scattering contribution was not performed to date and even its origin in nanocrystalline HPT Ni remains unclear.
The purpose of the present study is (i) to quantify the 'anomalous' magnetoelastic anisotropy previously suggested in pure HPT Fe and (ii) to further contribute to the understanding of the role played by the microstructure in the observation of higher-order contributions to the magnetic SANS cross section of HPT materials. In particular, we focus our analysis on the grain-size dependence of the structural and magnetic properties of nanocrystalline HPT Fe using a combination of standard techniques (i.e. X-ray diffraction, electron backscattered diffraction and magnetometry) with unpolarized magnetic SANS. The latter method is very powerful for obtaining bulk-averaged information on mesoscale structural and magnetic inhomogeneities; more specifically, magnetic SANS provides information about the variation of the magnetization vector field on a length scale of about 1-300 nm (Mü hlbauer et al., 2019;. For instance, magnetic SANS was previously employed to probe the spin structure of Heusler alloys (Runov et al., 2006;Bhatti et al., 2012;El-Khatib et al., 2019;McCalla et al., 2021;Bersweiler et al., 2022b), magnetic nanocomposites (Ito et al., 2007;Bick et al., 2013) or more recently to unravel the magnetic softness of Fe-Ni-B alloy, a HiB-NANOPERM-type soft magnetic material (Bersweiler et al., 2022a).
The paper is organized as follows: Section 2 provides details about the HPT sample preparation, the structural and magnetic characterization, as well as the neutron experiments. Section 3 provides a brief overview of the main expressions for the unpolarized magnetic SANS cross section, the recently derived analytical expressions for the spin-misalignment SANS cross section of textured ferromagnets are displayed and the neutron-data analysis is sketched. Section 4 presents and discusses the experimental results, and Section 5 summarizes the main findings of this study.

Experimental
The procedure for the preparation of the HPT Fe samples used in this paper is similar to the one described by Adachi et al. (2018). Sheets of Fe were cut into circular disks with a diameter of 20 mm. Prior to the HPT process, the samples were annealed at 1073 K for 1 h to homogenize the microstructure. The HPT process was then conducted under a compressive pressure of 5 GPa for N = 10 turns and with a rotation speed of 0.2 rpm. Subsequently, the disk-shaped samples were polished to remove surface roughness, oxides and possible impurities and were annealed at 473, 673 and 1273 K for 1 h; these are typical annealing temperatures for HTP Fe (Todaka et al., 2008;Shugaev et al., 2022). The final thickness of all the HPT samples was 0.36 mm. As previously reported, this specific approach of combining the HPT technique with an annealing process is a very effective way to easily obtain ultra-fine-grained Fe samples with various grain sizes (Adachi et al., 2018). In the following, the samples will be labeled as HPT + T a , where T a denotes the annealing temperature. Disks of non-deformed (nd) Fe with a diameter of 20 mm and a thickness of 0.88 mm were also prepared for comparison. Note that the thickness difference between the HPT + T a samples and the nd Fe sample results from the polishing after the HPT process and has no influence on the magnetic SANS results discussed below.
The microstructure of the HPT and nd Fe samples was characterized by wide-angle X-ray diffraction (XRD) using a Bruker D8 DISCOVER diffractometer in Bragg-Brentano geometry (Cu K radiation) and by electron backscattered diffraction (EBSD) using a Schottky field emission scanning electron microscope (SU5000, Hitachi High-Tech Corporation) with an electron backscatter detector (EDAX, Inc.). The EBSD patterns obtained were analyzed by the OIM analysis software (EDAX, USA). To fit with the requirements of the magnetometry and neutron experiments, rectangular-shaped samples with the surface area 10 Â 4 mm were cut. Roomtemperature magnetization curves for all samples were recorded using a Cryogenic Ltd vibrating sample magnetometer equipped with a 14 T superconducting magnet.
The neutron experiments were conducted at the instrument SANS-1 at the Swiss Spallation Neutron Source at the Paul Scherrer Institute, Switzerland. Fig. 1 is a sketch of the experimental SANS setup used for this study. The measurements were carried out using an unpolarized incident neutron beam with a mean wavelength of = 6.0 Å and a wavelength broadening of Á/ = 10% (full width at half-maximum). Sample-to-detector distances (L SD ) of 18 and 4.5 m were chosen to cover a q range of about 0.033 nm À1 q 0.94 nm À1 [q = (4/)/sin(É/2), where É is the scattering angle and is the wavelength of the incident radiation]. A magnetic field H 0 was applied perpendicular to the incident neutron beam (H 0 ? k 0 ). We would like to emphasize that, in this geometry, H 0 is perpendicular to the texture axis s that is induced by the HPT process (H 0 ? s). Neutron data were recorded by decreasing the field from the maximum field available of 8.0 T down to 0.2 T, following the magnetization curve (see Fig. 3). Within this field range, the neutron trans-  values were larger than 90%, indicating negligible multiple-scattering contributions. The neutron data reduction (corrections for background scattering and sample transmission) was conducted using the GRASP software package (Dewhurst, 2023).

Micromagnetic SANS theory
Based on the micromagnetic SANS theory for inhomogeneous ferromagnets (Honecker & Michels, 2013), the elastic total (nuclear + magnetic) unpolarized SANS cross section dAE/d at momentum-transfer vector q can be formally written as where, for the perpendicular scattering geometry (H 0 ? k 0 ), the first term represents the residual SANS cross section, which is measured at complete magnetic saturation, and the second term corresponds to the purely magnetic SANS cross section, which disappears at saturation. In the above expressions, V is the scattering volume, b H = 2.91 Â 10 8 A À1 m À1 is the atomic magnetic scattering length in the small-angle regime,Ñ NðqÞ and M MðqÞ ¼ ½M M x ðqÞ;M M y ðqÞ;M M z ðqÞ represent the Fourier transforms of the nuclear scattering length density N(r) and of the magnetization vector field M(r), respectively, and '*' denotes the complex conjugated quantities. M S (q) denotes the Fourier transform of the saturation magnetization profile M S (r).
In the neutron data analysis (see Section 4), to experimentally access the 2D dAE mag /d, we subtracted the 2D total SANS cross section dAE/d measured at the largest field of 8 T (approach-to-saturation regime; compare Fig. 3) from those measured at lower fields. This specific subtraction procedure eliminates the nuclear SANS contribution / jÑ Nj 2 , which is field independent, and therefore leads to the following expression for dAE mag /d: where Á represents the differences of the Fourier components at the two selected fields (low field À highest field).
In the case of the approach-to-saturation regime, wherẽ M M z ðqÞ 'M M S ðqÞ, the second-order micromagnetic SANS theory rigorously predicts thatM M z 2 ÀM M S 2 ¼ 0, so that the fourth term ÁM M z 2 sin 2 ðÞ in equation (4) disappears. Using the analytical results for the magnetization Fourier components of Honecker & Michels (2013) and Zaporozhets et al. (2022), it becomes possible to re-write equation (4) as where S H R H and S M R M correspond to the magnetic scattering contributions due to perturbing magnetic anisotropy fields and magnetostatic fields, respectively. In the particular case of an inhomogeneous polycrystalline ferromagnet with a global uniaxial anisotropy (magnetic texture), and when the texture axis is parallel to the incident neutron beam (as in our study), the anisotropy-field scattering function S H (q), the scattering function of the longitudinal magnetization S M (q), and the corresponding micromagnetic response functions R H (q, ) and R M (q, ) are given as follows (Zaporozhets et al., 2022): R M ðq; Þ ¼ 2 sin 2 ðÞ cos 2 ðÞ h q þ sin 2 ðÞ þ sin 2 ðÞ cos 4 ðÞ where Q ¼ 2K 0 =ð 0 M 2 0 Þ is the global anisotropy quality factor and K 0 (in J m À3 ) denotes the strength of the effective anisotropy, hĨ I 2 K i corresponds to the average of the squared Sketch of the magnetic SANS geometry. The momentum-transfer vector q corresponds to the difference between the wavevectors of the incident (k 0 ) and scattered (k 1 ) neutrons, i.e. q = k 0 À k 1 . The magnetic field H 0 is applied perpendicular to the incident neutron beam, i.e. H 0 k e z ? k 0 . The vector s represents the texture-axis direction, which in this study is parallel to the incident neutron beam, i.e. k 0 . The azimuthal angle characterizes the angular anisotropy of the recorded scattering pattern on the 2D detector. L SD corresponds to the sample-to-detector distance. For small-angle scattering, the component of the scattering vector along the incident neutron beam, here q x , is generally much smaller than the other two components q y and q z , so that only correlations in the plane perpendicular to the incoming neutron beam are probed.
Fourier image of the anisotropy constant fluctuations, hĨ I 2 M i is the corresponding quantity for the saturation magnetization fluctuations (assumed to be of the same order of magnitude as hĨ I 2 K i), M 0 = hM S (r)i represents the average saturation magnetization, and h q ¼ h þ L 2 0 q 2 is a dimensionless quantity related to the dimensionless applied-field parameter h = H 0 / M 0 and to the exchange (5)-(9) have already been successfully used to fit the angular dependence of the spin-misalignment SANS cross section of nanocrystalline HPT Ni, allowing one to estimate the global uniaxial anisotropy constant (Zaporozhets et al., 2022). Fig. 2(a) shows wide-angle X-ray diffraction data in the region of the (110) reflection for the annealed HPT Fe samples and for the nd Fe reference. We can see that although the (110) diffraction peak for HPT + 473 K is broad and not separable, the K 1 and K 2 lines for the (110) Fe reflection start to be resolved for HPT + 673 K and HPT + 1273 K. These features are indicative of an increase of the average crystallite size after the annealing process in our HPT samples. For the estimation of the average crystallite size D XRD , the Le Bail fit method implemented in the open-source software suite FullProf (Rodríguez-Carvajal, 1993) was used (considering the space group Im3m). The best-fit values are summarized in Table 1. The two following features are observed: (i) an ultra-finegrained microstructure with D XRD < 100 nm is obtained; and (ii) D XRD increases with increasing annealing temperature T a up to 673 K, before it slightly decreases again at 1273 K. Fig.  2(b) presents the inverse pole figure maps of HPT Fe and nd Fe obtained using EBSD. The estimated average grain sizes D EBSD are summarized in Table 1. Note that here the term 'grain' refers to an aggregate of several crystallites with the same (or nearly the same) crystallographic orientation. The individual crystallites cannot clearly be observed in Fig. 2(b), probably because of the resolution limit of the EBSD analysis.

Results and discussion
In this way, the rapid grain growth with increasing T a becomes more clearly visible. Furthermore, the difference between D XRD and D EBSD (see Table 1) suggests that the number of crystallites in a grain increases significantly with increasing T a . This result is related to recrystallization and subsequent grain growth, which occur for annealing temperatures higher than 473 K. As discussed in many publications, modified and enhanced recrystallization by HPT straining is the driving force of the grain-size dependence with the annealing temperature in HPT Fe (Voronova et al., 2007;Oberdorfer et al., 2010). Moreover, it is well known that the recrystallization process also gives rise to a recrystallization texture of an axial h110i type in HPT Fe (Shugaev et al., 2022). In Fig. 2(b), for HPT + 673 K, this is clearly seen as the development of {110} planes. The smaller population of the grains probably makes the observation of this texture more difficult for HPT + 1273 K. Fig. 3 displays the magnetization curves of HPT Fe and nd Fe on a semi-logarithmic scale. From the analysis of the magnetization curves, we estimated the saturation magnetizations M 0 (see Table 1). As shown in Fig. 3    display typical examples of the experimental 2D total SANS cross sections dAE/d of the HPT + 473 K Fe sample at the selected fields of 8 and 0.2 T, respectively. As can be seen, at 0 H 0 = 8 T (near saturation), dAE/d exhibits a nearly isotropic pattern, which suggests the dominance of isotropic nuclear scattering. According to the magnetization curves [see Fig. 3], the sample is nearly magnetically saturated at a magnetic field of 8 T. By reducing H 0 , the pattern at the larger (smaller) momentum transfers becomes slightly elongated perpendicular (parallel) to H 0 , suggesting a more complex magnetization structure corresponding to the other terms in the magnetic SANS cross section [equation (4)]. Fig. 4(c) shows the corresponding 2D purely magnetic SANS cross section dAE mag /d, obtained from the subtraction of (a) and (b). As is seen, the vertical and horizontal elongations remain in dAE mag /d in the respective q range after the subtraction procedure [compare also Figs. 4(d) and 4(e)]. The horizontal elongation can be attributed to the spin-misalignment scattering due to long-wavelength transversal magnetization fluctuations [compare the ÁM M y 2 cos 2 ðÞ term in equation (4)]. As recently discussed by Oba et al. (2021) for ultra-fine-grained Ni prepared by HPT, the vertical elongation might be related to the higher-order terms in the expression for dAE mag /d [compare the ÁjM M z j 2 sin 2 ðÞ term in equation (4) Upper-right branch of the normalized magnetizations of HPT Fe and nd Fe measured at room temperature and between + 14 T and + 0.05 T (semi-logarithmic scale). Gray area: approach-to-saturation regime defined as M/M 0 ! 90%. The magnetization curves have been normalized by the saturation magnetizations M 0 , which were estimated for each sample from the linear regression M(1/H 0 ) in the high-field regime (i.e. 0 H 0 2 10 T À 14 T). The corresponding values are given in Table 1.

Figure 4
Experimental 2D elastic total (nuclear + magnetic) unpolarized SANS cross section dAE/d of the HPT + 473 K Fe sample measured at the selected fields of (a) 8 T and ( studied a series of as-cast, aged and mechanically deformed bulk metallic glasses. The relevance of the higher-order terms in the magnetic SANS cross section will be discussed more in detail later in the manuscript. Fig. 5 presents the magnetic-field dependence of the azimuthally averaged total SANS cross section dAE/d. As can be seen, at 0 H 0 = 8 T (near saturation), dAE/d can be well described by an asymptotic power-law exponent dAE/d / q À4 (at the smallest and intermediate q range), which is expected in the Porod regime and therefore suggests that dAE/d at 8 T is a good approximation of the residual SANS cross section dAE res /d. Compared with the cross section at 8 T, for HPT + 473 K and HPT + 673 K, the respective SANS profile at 1.0 T exhibits a (more or less pronounced) shoulder, similar to what has been reported in previous SANS studies of HPT ferromagnets (Oba et al., 2020;Bersweiler et al., 2021). With decreasing field, the scattering intensity increases by more than two orders of magnitude around the shoulder (q = 0.06 -0.3 nm À1 ). Since the nuclear scattering is field-independent, the origin of the strong field dependence observed in HPT + 473 K and HPT + 673 K can be attributed to the spin misalignment, which is induced via HPT straining and persists in fields between 0.2 and 1.0 T. By contrast, for HPT + 1273 K and nd Fe, the q dependence of dAE/d becomes more similar to that obtained for dAE res /d / q À4 (at the smallest momentum transfers) and weakly field dependent. This suggests the presence of larger (nuclear + magnetic) correlation lengths in both samples (lying outside of the measured q range) and that the microstructures before deformation substantially reappear in HPT + 1273 K due to annealing. This hypothesis is supported by the rapid grain growth observed with increasing annealing temperature [see Fig. 2(b)]. Fig. 6 displays the azimuthally averaged purely magnetic SANS cross section dAE mag /d. The magnitudes of the dAE mag / d are of the same order as the dAE/d shown in Fig. 5, therefore supporting the notion of dominant spin-misalignment scattering in HPT Fe. As previously suggested, it may originate from induced structural defects, which act as a source of effective magnetoelastic anisotropy field (Oba et al., 2020). To estimate the anisotropy strength of the inhomogeneities, we have fitted the magnetic-field dependence of dAE mag /d according to equation (5), using the scattering and response functions given by equations (6)-(9) (solid lines in Fig. 6). More precisely, by taking the experimental value of the average saturation magnetization 0 M 0 = 2.1 T (of the HPT 473 K and HPT 673 K samples) and the estimated value of the exchange length L 0 = 3.5 nm (assuming A ex = 21 pJ m À1 ), only the global anisotropy factor Q is unknown in equations (6)-(9). The value for L 0 (A ex ) controls the global rate of decay with q of all the cross sections. Because of the linearity of equation (5) in R H and R M , one can obtain the values of S H and S M at each q value by performing a (weighted) leastsquare fit of dAE mag /d measured at the different selected fields. 1 The total least-square error of the fits was then numerically minimized to obtain the value of Q. The corresponding values for the global uniaxial anisotropy K 0 ¼ 0 M 2 0 Q=2 obtained for HPT + 473 K and HPT + 673 K are listed in Table 1. Note that the texture axis is assumed to be parallel to the incident neutron beam (i.e. along the HPT strain direction). As can be seen, the uniaxial anisotropy Magnetic field dependence of the (over 2) azimuthally averaged total (nuclear + magnetic) SANS cross section dAE/d of HPT Fe and nd Fe (log-log scales). Black solid lines: asymptotic power law dAE/d / q À4 .

Figure 6
Magnetic field dependence of the (over 2) azimuthally averaged purely magnetic SANS cross section dAE mag /d of HPT Fe and nd Fe (log-log scales). The dAE mag /d curves have been obtained from the data plotted in Fig. 5 by subtracting the dAE/d at 8.0 T from the dAE/d at lower fields. Black solid lines: fits by equation (5) using the scattering and response functions defined by equations (6)-(9), which are valid in the approachto-saturation regime. Note that we have restricted our fit analysis to the HPT + 473 K and HPT + 673 K datasets, for which the magnetic correlation lengths can be spatially resolved within the available experimental q range [i.e. for neutron data which do not exhibit a power-law behavior of dAE mag /d / q Àn with n varying from $3.5 (4.1) to $3.9 (3.4) for HPT Fe (nd Fe)]. values obtained are slightly larger than the magnetocrystalline anisotropy value of 0.048 MJ m À3 reported for bulk Fe (Graham, 1958). Therefore, this result strongly supports the existence of an induced magnetoelastic anisotropy in pure Fe prepared by HPT straining, as previously suggested by Oba et al. (2020).
Moreover, as noticed in Fig. 6, the quality of the fits is reasonable with the largest deviations observable for the HPT + 673 K sample at around q = 0.15 nm À1 . The (small) discrepancy between the experimental neutron data and the fits using equation (5) highlights the limits of the micromagnetic SANS theory for highly inhomogeneous ferromagnets (i.e. this study). More specifically, in the approach-to-saturation regime, the second-order micromagnetic SANS theory rigorously predicts that ÁjM M z j 2 ¼ 0, so that the fourth term ÁjM M z j 2 sin 2 ðÞ in equation (4) disappears (see Section 3). However, as already discussed in previous studies (Metlov & Michels, 2015;Metlov et al., 2020;Oba et al., 2021), it is entirely possible that for highly inhomogeneous ferromagnets, the higher-order terms become non-negligible, therefore breaking this property. Generally, the higher-order terms are masked by the lower-order terms. But, using unpolarized magnetic SANS, the field dependence of the higher-order contribution can be experimentally highlighted by considering the following combination of radially averaged SANS cross sections (Metlov & Michels, 2015): 2 Inserting equations (5)-(9) into equation (10) shows that, for Q = 0, this combination of cross-section values is exactly zero (in second order). Non-zero ÁAE mag indicates the presence of higher-order scattering contributions. Fig. 7(a) shows the higher-order scattering contribution ÁAE mag in HPT Fe and nd Fe at the selected field of 200 mT. As is seen, the magnitude and sign of ÁAE mag of these samples are strongly affected by the annealing temperature, which is further supported by the results for the field dependences of the individual ÁAE mag in Fig. 7(b). For HPT + 423 K and at 200 mT, ÁAE mag exhibits a local minimum at an intermediate q range, whereas for HPT + 673 K, ÁAE mag becomes strictly negative and small; for HPT + 1273 K, ÁAE mag ' 0. This result contrasts with the one obtained for nd Fe, for which ÁAE mag is strictly positive over the full q range. At the smallest momentum transfers and for the HPT + 423 K sample at 200 mT, the ratio of ÁAE mag to dAE mag /d is about 3À4%. The negative sign of ÁAE mag for HPT + 423 K can be qualitatively explained considering the micromagnetic SANS theory for a weakly inhomogeneous magnetic material (Metlov & Michels, 2015), which predicts that, in the case of anisotropy inhomogeneities with a very small amplitude, ÁAE mag may have negative values at an intermediate q range. Further extension of the higher-order effect theory is required to fully explain the strictly negative values of ÁAE mag over the whole q range in HPT + 673 K; a negative ÁAE mag was already reported for the case of HPT Ni . Moreover, supported by the results of our structural characterization (XRD and EBSD), it appears clear that the variation in the magnitude of ÁAE mag in the HPT Fe samples is related to a change in their microstructure. Here, the reduction in the magnitude of ÁAE mag observed for the HPT Fe samples with increasing T a can be attributed to a reduction in the density of defects that originate from the recovery and recrystallization process, which occurs at annealing temperatures larger than 473 K. Note that the shape of the inclusions (defects) might also play a significant role in the magnitude of ÁAE mag . As theoretically explored by Schlö mann (1971)  materials, the higher-order contribution is found to be small for spherical inclusions and approximatively six times larger for layer-like inclusions. However, since the HPT process does not allow for the control of the shape of the grains in a material, this conjecture of Schlö mann is difficult to verify or falsify. Fig. 7(b) displays the magnetic-field dependence of ÁAE mag in HPT Fe and nd Fe. As expected, the decrease of the magnitude of ÁAE mag with increasing H 0 can be attributed to the fact that the transversal Fourier componentsM M x ðqÞ and M M x ðqÞ ! 0 and that the longitudinal componentM M z ðqÞ ! M M S ðqÞ, which therefore implies ÁjM M z ðqÞj 2 ! 0 in equations (3) and (4).

Conclusions
We employed a combination of standard characterization techniques (XRD, EBSD, magnetometry) with a more advanced neutron method (magnetic SANS) to investigate the induced magnetoelastic anisotropy and the role played by the microstructure in the observation of higher-order scattering contributions in nanocrystalline Fe prepared by high-pressure torsion associated with a post-annealing process. The structural characterization confirms the formation of ultra-finegrained pure Fe with different grain sizes by varying the subsequent annealing temperature. The fits of the neutron data to the micromagnetic SANS theory extended for textured ferromagnets yield anisotropy values of $0.059 and $0.073 MJ m À3 in ultra-fine-grained Fe with an average crystallite size of $48 and $98 nm, respectively. These values are larger than the magnetocrystalline anisotropy values for bulk Fe and strongly suggest the existence of an induced magnetoelastic anisotropy in ultra-fine-grained Fe prepared by HPT straining. Furthermore, the analysis of the magnetic field-dependent unpolarized SANS data unambiguously reveals the presence of non-negligible higher-order scattering contributions in the HPT + T a samples. While the sign of the higher-order contribution might be related to the amplitude of the anisotropy inhomogeneities, as predicted by Metlov & Michels (2015), its magnitude appears clearly correlated to the changes in the microstructure (density and/or shape of the defects) induced by HPT and the subsequent annealing process. Finally, these results point out the necessity to consider the higher-order terms in the expression of the SANS cross section in the case of highly inhomogeneous ferromagnets.